Cantilever
A beam clamped at one end and loaded at free end is called a cantilever . Depression at the free end of a cantilever is given by δ = wl 3 / 3YIG
where , w = load , 1 = length of the cantilever ,
y = Young ’ s modulus of elasticity , and IG = geometrical moment of inertia . For a beam of rectangular cross-section having breadth b and thickness d . IG = bd 3 / 12 For a beam of circular cross-section area having radius r ,
IG = π r 4 / 4
Beam Supported at Two Ends and Loaded at the Middle
Depression at middle δ = wl 3 / 48YIG Torsion of a Cylinder